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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习4.1-4.2 - 复级数的基本性质、幂级数 }
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\date{2024 年 4 月 29 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
考察下述复级数的敛散性： 
$$ 
(1)\sum\limits_{n=1}^{\infty} \left(\frac{1}{n} + \frac{i}{2^n} \right);  \hspace{1cm} 
(2) \sum\limits_{n=1}^{\infty} \frac{i^n}{n}; \hspace{1cm} 
(3) \sum\limits_{n=1}^{\infty} \frac{(3+5i)^n}{n!}; \hspace{1cm} 
(4) \sum\limits_{n=1}^{\infty} \left(\frac{1+5i}{2}\right)^n. 
%\hspace{1cm} (2) \sum\limits_{n=1}^{\infty} \frac{1}{n^z}; 
%\hspace{1cm} (3) \sum\limits_{n=1}^{\infty} (z^n-z^{n-1}). 
$$

\vspace{0.2cm}

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\item  %Problem 02
证明级数 $1+z+z^2+\cdots+z^n+\cdots$ 在闭圆 $|z|\le r\,\, (r<1)$ 上一致收敛。

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\item  %Problem 03 
（魏尔斯特拉斯定理）设函数 $f_n(z)\, (n=1,2,\cdots)$ 在区域 $D$ 内解析，设函数项级数 $\sum\limits_{n=1}^{\infty} f_n(z)$ 在区域 $D$ 内内闭一致收敛于函数 $f(z)$. 证明函数 $f(z)$ 在区域 $D$ 内解析，且有 $$ f'(z)=\sum\limits_{n=1}^{\infty} f_n'(z). $$

\vspace{0.2cm}

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\item  %Problem 04 
（蒙泰尔定理）设复函数序列 $\{f_n(z)\}$ 在区域 $D$ 内解析，并且在 $D$ 内内闭一致有界。则存在子序列 $\{f_{n_k}(z)\}$ 在 $D$ 内内闭一致收敛，并且这个子序列的极限函数在区域 $D$ 内解析。

\vspace{0.2cm}

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\item  %Problem 05
（阿贝尔定理）设幂级数 $\sum\limits_{n=0}^{\infty} c_n(z-a)^n$ 在某点 $z_1\neq a$ 收敛，则它必在圆
$K: |z-a|<|z_1-a|$ 内绝对收敛，且内闭一致收敛。

\vspace{0.2cm}

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\item  %Problem 06
（柯西-阿达马公式）设幂级数 $\sum\limits_{n=0}^{\infty} c_n(z-a)^n$ 的系数 $c_n$ 满足下述条件之一，
$$
\ell = \lim\limits_{n\to\infty} \lvert \frac{c_{n+1}}{c_n} \rvert, \hspace{0.5cm}
\ell = \lim\limits_{n\to\infty} \sqrt[n]{|c_n|}, \hspace{0.5cm}
\ell = \varlimsup\limits_{n\to\infty} \sqrt[n]{|c_n|}, 
$$
则这个幂级数的收敛半径为 $R=1/\ell$. 

\vspace{0.2cm}

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\item  %Problem 07
求下述幂级数的收敛半径，
$$
(1) \sum\limits_{n=1}^{\infty} \frac{z^n}{n^2}; \hspace{0.5cm} 
(2) \sum\limits_{n=1}^{\infty} \cos(in)(z-1)^n; \hspace{0.5cm} 
(3) \sum\limits_{n=1}^{\infty} n! z^n; \hspace{0.5cm} 
(4) \sum\limits_{n=1}^{\infty} (3+4i)^n(z-i)^{2n}. 
$$


\vspace{0.2cm}

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\item  %Problem 08
证明幂级数 $\sum\limits_{n=0}^{\infty} c_n(z-a)^n$ 的和函数 $f(z)$ 在其收敛圆 $K: |z-a|<R$ 内是解析，
且有求导公式 $$f'(z) = \sum\limits_{n=1}^{\infty} nc_n(z-a)^{n-1}. $$

\vspace{0.2cm}

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\item  %Problem 09
求下述幂级数的收敛半径，
$$
(1) \sum\limits_{n=1}^{\infty} \frac{z^n}{n}; \hspace{1cm} 
(2) \sum\limits_{n=1}^{\infty} \frac{nz^n}{2^n}; \hspace{1cm} 
(3) \sum\limits_{n=1}^{\infty} n^nz^n. 
$$

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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